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A set of seismic observations which all sample the same structure in the same way should have zero variance. This is naturally the case if all sources are in the same place, and the data are recorded by the same station. If sources and/or receivers are not in the same place, but close to one another, variance will generally be nonzero, but small. Variance might become large if the sampled region of the Earth contains heterogeneities whose spatial wavelength is comparable to the distances between sources and between receivers (and thus between the corresponding ray paths). The travel-time variance of a “bundle” of seismic rays thus reflects the degree of complexity of the sampled region of the medium. We apply this simple principle to real seismic databases, attempting to constrain the spherical harmonic spectrum of Earth’s structure without having to derive a tomographic model. This results in a reduction of the dimensionality of the solution space, and hence of computational costs. This approach allows to constrain the statistical properties, rather than exact geographic locations of structural features; knowing the statistics of Earth’s structure is most valuable for many fundamental geodynamic questions. We follow an earlier study by Gudmundsson et al.  to find an approximate analytical relationship between averaged variance and harmonic spectrum; this allows us to determine the latter from a measurement of the former via a linear least squares inversion. Our analysis shows that the variance of ray bundles associated with large geographic extent of source/receiver bins is sensitive to low-degree spectral power, and vice-versa for small bins/high harmonic degrees. The method is accordingly ineffective at very low harmonic degrees, associated with an inherently limited number of source-receiver bins. We conduct a suite of inversions of both real and synthetic seismic data sets to evaluate the resolving power of our algorithm, and attempt to identify a range of harmonic degrees where the method is robust. Our results indicate that the resolution of the Earth’s spectrum afforded by the method presented here is inferior to that of classical tomography.
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